Heisenberg's Equality of Inequivalents Problem

In his book Physics and Philosophy', Heisenberg suggested that the quantum world is one of ``potentialities or possibilities'' and that the classical realm is one of ``things or facts''. After ascertaining that his categories most naturally have the structure of ontological equivalence classes, we show that they cannot be seriously introduced into the quantum formalism without rendering it incoherent, as under some circumstances the formalism permits an equality between members of distinct equivalence classes. This is labeled the equality of inequivalents problem’. Three possible reactions are discussed: First, one could deny the problem by challenging some of the assumptions that underlie its formulation; second, one could accept the problem and take it to be grounds for dismissing Heisenberg’s or similar distinctions altogether; third, one could accept the problem but take its source to be not Heisenberg’s distinction but the quantum formalism. A plausibility argument in support of the third reaction is given by analogy: it is shown that axiomatic probability suffers from a similar problem if one wishes to seriously distinguish between things which are possibilities and things which are not. An enriched axiomatization of probability is proposed which captures the concept of probability as a measure over possibilities and thereby overcomes the equality of inequivalents problem in axiomatic probability.